1、Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NASA SP-160 PL Ohio State University Columbus, Ohio ScientGc and Technical In formation Dzvzszon OFFICE OF UTILIZATION 1969 NATIONAL AERONAUTICS AND SPACE ANSTRATIQN Washington, D.C. Provided by IHSNot
2、for ResaleNo reproduction or networking permitted without license from IHS-,-,-For sale by the Superintendent of Documents US. Government Printing Otfce, Washington, D.C. 20402 Price $3.50 (paper cover) Library of Congress Catalog Card Number 6742660 Provided by IHSNot for ResaleNo reproduction or n
3、etworking permitted without license from IHS-,-,-r of books, papers, and reports e). The present monograph attempts to shapes of free vibration of plates would be provided for the design or develop- ults would be provided for the researcher engineer can develop a qualitative understanding of plate v
4、ibrational behavior. For the afore at least two approximate formulas are given for estimates inally, the mathematical techniques used in the literature to solve the problem or related ones are pointed out in case more accurate results are needed, It is my hope that this monograph will reduce duplica
5、tion of research effort in plate vibrations in the future (a very pointed example is that of the square plate clamped all around). In addition, the researcher is provided accurate numerical results for the testing of new methods (this is the reason that results ght significant figures in some cases)
6、. Finally, it is hoped that ive added perspective to the merits and complexities of applying analytical techniques to eigenvalue problems. IIE Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-N VIBRATION OF PLATES Gaps in knowledge are made implicitly
7、 obvious by examining this work. For example, analytical results have been found for a clamped elliptical plate, and experimental results for the free case, but no results whatsoever have been found for the simply supported case. The scope of this study was limited by several considerations. Only th
8、e analytical results from plate theories were considered; that is, the governing equations are two-dimensional, not three-dimenpional. Materials were re- stricted to those which are linearly elastic. Structures were not included in the study; for example, a rectangular plate supported by one or more
9、 edge beams was considered to be a structure. The primary logical division of this work is by the complexity of the governing differentia1 equations. Thus, the first eight chapters deal with the simplest “c1assicaI theory” of plates. The next three chapters introduce the complications of anisotropy,
10、 in-plane force, and variable thickness. Other complications are discussed in the twelfth chapter. The first subdivision is by geometrical shape; that is, circles, ellipses, rectangles, parallelograms, and so forth. Further subdivision accounts for holes, boundary conditions, added masses or springs
11、, and so forth. It is presupposed that the user of this monograph will have at least an elementary understanding of plate theory. In order to increase understanding and to define notation and assumptions more clearly, a reasonably rigorous derivation of the plate equations is made in the appendix. S
12、ome statements about the format of presentation will be useful in under- standing this work. It will be seen that the majority of results available are for the natural frequencies of free vibration and quite often only the funda- mental (lowest) frequency. Patterns showing node lines are frequently
13、available for the higher modes. Mode shapes (deflection surfaces in two dimensions) are usually not completely specified in the literature. It should be remarked here that the mode shapes (eigenfunctions) cannot be completely determined until the frequencies (eigenvalues) are found. The mode shapes
14、are generally known less accurately than the frequencies. Virtually no one in the literature evaluates the bending stresses due to a unit amplitude of motion. This information is obviously important, particu- larly for fatigue studies. The lack of results is undoubtedly due to the fact that the stre
15、sses must be obtained from second derivatives of the mode shapes. Not only does this require additional computational work, but also the mode shapes usually are not known with sufficient accuracy to give meaningful resdts for stresses. Frequency data were converted to the angular frequency o (radian
16、slunit time) or to a corresponding nondimensional frequency parameter, where possible. Almost always the number of significant figures was kept the same as that in the original publication. In no case were significant figures added. In some few cases the number of significant figures was reduced bec
17、ause the accuracy of the calculations in the publication did not justify the numbers gven. Curves were not replotted but were photographically enlarged and traced to maximize accuracy. Quite often, when they are available, both tabular and graphical results are given for a problem. Tabular results a
18、re particularly important for measuring the accuracy of an analytical method, whereas curves are valuable for interpolation, extrapolation, and qualitative studies. In some cases many sets of results are given for the same problem. Provided by IHSNot for ResaleNo reproduction or networking permitted
19、 without license from IHS-,-,-PREFACE V In these cases each set was derived by a different theoretical or experimental technique; this permits a comparison of techniques. Two of the major goals of the project were accuracy and completeness. Some of the efforts made to maintain accuracy have been des
20、cribed in the eteness of results published through Writing of the manuscript began in the he well-known abstracting journals, re used in order to procure pertinent references were obtained from the e already procured. Approximately d. out the world who were ns. These letters listed r copies of any o
21、thers which come to possess a reasonably com- plete set of literature in the field of plate vibrations. However, in spite of this, I am convinced that some significant publications are not included, particularly some which are known to exist but have been thus far unobtainable, especially books by S
22、oviet researchers. In light of the preceding paragraph, I expect-indeed, hope-to receive considerable valuable criticism pointing out errors or omissions. In addition, I would appreciate receiving copies of recent or forthcoming publications and reports which are pertinent. It is my intention to wri
23、te a supplement to this volume after a few years have elapsed; such a document will correct any major mistakes or omissions in this work and will report on further advances in the field. For historical record and recognition it should be pointed out that, ap- proximately 6 months after this project
24、began, I discovered a notable work entitled “Free Vibrations of Plates and Shells,” by V. S. Gontkevich, published (in Russian) in 1964. A subsequent complete translation into English was made under the sponsorship of the Lockheed Missiles C, C:, S, S:, F, F:, G, and Gi are constants of integration;
25、 and p=P=w .JJD (1.25) he complete solution to equation (1.4) is then = Wl+ w2 (1.26) For a solid region containing the origin, regdar- ity conditions require that half of the terms in equations (1.24) be discarded, and the complete solution becomes : Y X FIGURE 1.2.-Elliptical coordinate system. Pr
26、ovided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 VIBRATION OF PLATES 1.3 RECTANGULAR COORDINATES shown in figure 1.3. The rectangular coordinates of a point P are Y FIGURE 1.3.-Rectangular coordinate system. .3.6 Classical Equations The Laplacian oper
27、ator in rectangular co- ordinates is (1.28) ending and twisting moments are related to the displacements by b2W Mzu= -D (1 -v) - dy earing forces are given and the Kelvin-Kirchhoff edge reactions are The strain energy of bending and tuisting of a piate expressed in rectangular coordinates is dA (1.3
28、2) where dA=dx dy. 1.3.2 Solutions General solutions to equation (1.4) in rec- tangular coordinates may be obtained by assuming Fourier series in one of the variables, say x; that is, Substituting equation (1.33) into equation (1 .S yields d2Ym1 + (k2-a2)Pm1 =O (1.34) dY2 dY2 d2Ym2 - (kq d)Y,=O and
29、two similar equations for y*, With the assumption that k2a2, solutions to equations (1.34) are well known as Ym,=A,sin Jk%2y+B, cosJk2-2y Ym2=Cm sinhJWy+D, coshy (1.35) where A, . . ., D, are arbitrary coefficients determining the mode shape and are obtained from the boundary conditions. If k2 - at2
30、h (1.41) The edge reactions are (ref. where B= (?r/2) -a. 1.7) : -2 cos p ( 1.42) (I-Y) d2W . b2W -sin a at The strain energy of bending and twisting of M,=-D- - COSa (bbQ 8 plate expressed in skew coordinates is (1.40) 1.4.2 Sofutions -9 P I I I I There are no known general solutions to equation (1
31、.4) in skew coordinates which allow a separation of variables. X * 1.1. MCLACHLAN, N.: essel Functions for Engineers. Oxford Eng. Sci. Ser., Oxford Univ. Press (London), 1948. FIGURE 1.4.-Skew coordinate system. Provided by IHSNot for ResaleNo reproduction or networking permitted without license fro
32、m IHS-,-,-6 VIBRATION OF PLATES 1.2. NASH, W. A.: Bending of an Elliptical Plate by J. Appl. Mech., vol. 17, no. 3, 1.3. GALERKIN, 3. G.: Berechnung der frei gelagerten ZAMM, Bd. 3, 1.4. CHENG, SHUN: Bending of an Elliptic Plate Under a Moment at the Center. Tech. Sum. Rept. No. 444, Math. Res. Cent
33、er, Univ. Wisconsin, Dec. 1963. Edge Loading. Sept. 1950, pp. 269-274. elliptischen Platte auf Biegung. 1923, pp. 113-117. 1.5. MCLACHLAN, N.: Theory and Application of Oxford Univ. Press (Lon- 1.6. MORLEY, L. S. D.: Skew Plates and Structures. 1.7. DMAN, S. T. A.: Studies of Boundary Value Problems
34、. Part 11, Characteristic Functions of Rectangular Plates. Proc. NR 24, Swedish Cement and Concrete Res. Inst., Roy. Inst. Tech. (Stockholm), 1955, pp. 7-62. Mathieu Functions. don), 1947. Macmillan Co., Inc., 1963. Provided by IHSNot for ResaleNo reproduction or networking permitted without license
35、 from IHS-,-,-Chapter 2 es 2.1 SOLID CIRCULAR PLATES When the origin of a polar coordinate system is taken to coincide with the center of the circular plate and plates having no internal holes are considered, the terms of equation (1.18) involving Yn(kT) and K,(kT) must be discarded in order to avoi
36、d infinite deflections and stresses at r=O. If the boundary conditions possess symmetry with respect to one or more diameters of the circle, then the terms involving sin ne are not needed. When these simplifications are employed, equation (1.18) becomes for a typical mode : Wn=AnJn(kr) +CJ,(kr) COS
37、ne (2.1) where it will be understood in what follows that n can take on all values from 0 to 03. The subscript n will also correspond to the number of nodal diameters. 4.1.1 Piates Ciamped All Around d around be a (see fig. 2.1). conditions are: Let the outside radius of the plate clamped The bounda
38、ry en equation (2.1) is substituted into equa- tions (2.2), the existence of a nontrivial solution yields the characteristic determinant (2.3) where X ka and the primes are used to indicate erentiation with respect to the argument, io this case kr. Using the recursion relationships (ref. 2.1) XJ(h=n
39、Jn(X-XJn,(X) K(X)=nIa(X) +uw+,(X) (2.4) FIGURE 2.1.-Clamped circular plate. and expanding equation (2.3) gives Jn(Xln+,(X+ln(X)Jn+I(x)=O (2.5) The eigenvalues h determining the frequencies w are the roots of equation (2.5). The Bessel functions are widely tabulated for small values of n. The Narvard
40、 tables (ref. 2.2) are avaabe for n =O (2.10) PGUBE Z.Z.-Sionply supported eireukr plaie. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-CIRCULAR PLATES 9 TABLE 2.2.-Radii of Nodal Circles p=r/ajor Clamped Circular Plate S 0 1.0 1.0 .379 .583 .255 .
41、688 .439 .191 .749 .550 .351 .153 .791 .625 .459 .293 .127 .822 .678 .535 .393 .251 -109 .844 .720 .593 .469 .344 .220 .096 1.0 1.0 1.0 1.0 1.0 1 1.0 1.0 .4899 .640 .350 .721 .497 .272 1.0 .767 .589 .407 .222 .807 .653 .499 .344 .188 .833 .699 .566 .432 .298 .163 .853 .735 .617 .499 .381 .263 .I44 1
42、.0 1.0 1.0 1.0 where the notation of the previous section is used. It has been shown (ref. 2.11) that equa- tions (2.10) lead to the frequency equation ts of equation (2.11) and radii of nodal es for v=0.3 are taken from reference 2.6 and presented in tables 2.3 and 2.4, respectively. son, in an ear
43、ly paper (ref. 2.12), and cott (ref. 2.11) gbe X=2.204 for v=0.25. Bodine (ref. 2.19) (see section entitled “Plates 808-337 0-7L-2 p for values of n of- 2 1.0 1.0 .559 .679 .414 .?46 .540 .330 .789 .620 .449 .274 1.0 1.0 . - - - - - - - - - - 3 1.0 1.0 .606 .708 .462 .?65 .574 .375 .803 .645 .488 .3
44、16 1.0 1. 0 _-_ TABLE 2.3.-Values of X2=wa2Jm for a Simply Supported Circular Plate; v=O.S A2 for values of n of- S I 2 13.94 48.51 102.80 176.84 25.65 70.14 134.33 218.24 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-VIBRATION OF PLATES 10 I 1 1 1
45、 .613 .726 .443 .787 .570 .348 1 TABLE 2.4.-Radii of NGM Circles p=rlafor a Simply Supported Circular Plate; v=0.3 p for values of n of- 8 I I 1 .736 .469 .204 1 1 1 .550 .692 .378 .765 .528 .288 t Supported on Circle of Arbitrary Radius” (2.1.7) gives X=2.228 for v=0.333. The mode shapes are most c
46、onveniently determined from the first of equations (2.10) by use of the roots of table 2.3; that is, (2.12) he procedure for determining the motion of a plate subjected to arbitrary initial displace- ment and velocity conditions is given in reference 2.7. The simply supported case is also solved in
47、reference 2.20. For more information concerning this pr lem, see section entitled “Simply Suppor Circular Plates (10.12). 9.3.3 Completely Free Plates Let the outside radius of free plate be a (see fig. 2.3). conditions are M,(Ct)=O V,(a)=o the completely The boundary (2.13) Using equations (1.11),
48、(1.121, (1.13, it has been shown (ref. 2.3) that equations (2.13) yield the frequency equation FIGURE 2.3.-Free circular plate. - x3G(x+(1-v)n2 IXJk-Jn(xl (2.14) x31h(X)-(1 -v)n2 AIh(A)-IJ,(x) It has also been shown (ref. 2.20) that, when An, one can replace equation (2.14) by the approximate formula n, J (A) IxZ+2(1-)n2jIIn(X)/1k(X)I-22X(1-) Jib- x2-2(1-v)n2 (2.15) According to reference 2.20, the roots of equation (2.14) are located between the zeroes of the functions Jk(X and Jn(X and the larger roots may be calculate expansion (2.16) wherem
